# A Geometry of ApproximationLocal Validity, Grothendieck Topologies and Rough Sets

A Geometry of Approximation: Local Validity, Grothendieck Topologies and Rough Sets Chapter 7 Local Validity, Grothendieck Topologies and Rough Sets 7.1 Representing Rough Sets The ﬁrst step is to represent rough sets. Thus, we now give the formal deﬁnition of a rough set and the formal deﬁnition of the decreasing representation of rough sets which was adopted in the Introduction. Deﬁnition 7.1.1. Given an Indiscernibility Space U, E, 1. Two sets X, Y ∈ ℘(U ) are called rough top equal, X 0 Y ,iﬀ (uE)(X)= (uE)(Y ). 2. Two sets X, Y ∈ ℘(U ) are called rough bottom equal, X∼Y ,iﬀ (lE)(X)= (lE)(Y ). 3. Two sets X, Y ∈ ℘(U ) are called rough equal, X ≈ Y ,iﬀ X 0 Y and X ∼ Y . 4. A set X ∈ ℘(U ) is called deﬁnable iﬀ X =(lE)(X)= (uE)(X). 5. A set X ∈ ℘(U ) is called undeﬁnable iﬀ (lE)(X)= ∅ and (uE)(X)= U . 6. Any equivalence class of subsets of U modulo the relation ≈ is called a rough set. 211 212 7 Local Validity, Grothendieck Topologies and Rough Sets We represent rough sets, with respect to an Indiscernibility Space U, E by means of ordered pairs of the form (uE)(X), (lE)(X),and http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Geometry of ApproximationLocal Validity, Grothendieck Topologies and Rough Sets

Part of the Trends in Logic Book Series (volume 27)
Editors: Pagliani, Piero; Chakraborty, Mihir
Springer Journals — Jan 1, 2008
26 pages      /lp/springer-journals/a-geometry-of-approximation-local-validity-grothendieck-topologies-and-6xcrxUmPOU
Publisher
Springer Netherlands
ISBN
978-1-4020-8621-2
Pages
211 –236
DOI
10.1007/978-1-4020-8622-9_7
Publisher site
See Chapter on Publisher Site

### Abstract

Chapter 7 Local Validity, Grothendieck Topologies and Rough Sets 7.1 Representing Rough Sets The ﬁrst step is to represent rough sets. Thus, we now give the formal deﬁnition of a rough set and the formal deﬁnition of the decreasing representation of rough sets which was adopted in the Introduction. Deﬁnition 7.1.1. Given an Indiscernibility Space U, E, 1. Two sets X, Y ∈ ℘(U ) are called rough top equal, X 0 Y ,iﬀ (uE)(X)= (uE)(Y ). 2. Two sets X, Y ∈ ℘(U ) are called rough bottom equal, X∼Y ,iﬀ (lE)(X)= (lE)(Y ). 3. Two sets X, Y ∈ ℘(U ) are called rough equal, X ≈ Y ,iﬀ X 0 Y and X ∼ Y . 4. A set X ∈ ℘(U ) is called deﬁnable iﬀ X =(lE)(X)= (uE)(X). 5. A set X ∈ ℘(U ) is called undeﬁnable iﬀ (lE)(X)= ∅ and (uE)(X)= U . 6. Any equivalence class of subsets of U modulo the relation ≈ is called a rough set. 211 212 7 Local Validity, Grothendieck Topologies and Rough Sets We represent rough sets, with respect to an Indiscernibility Space U, E by means of ordered pairs of the form (uE)(X), (lE)(X),and

Published: Jan 1, 2008